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We analyze the pion transition form factor using dispersion theory. We calculate the singly-virtual form factor in the time-like region based on data for the e⁺e⁻→3π cross section, generalizing previous studies on ω,ϕ→3π decays and γπ→ππ scattering, and verify our result by comparing to e⁺e⁻→π⁰γ data. We perform the analytic continuation to the space-like region, predicting the poorly-constrained space-like transition form factor below 1GeV, and extract the slope of the form factor at vanishing momentum transfer aπ=(30.7±0.6)×10⁻³. We derive the dispersive formalism necessary for the extension of these results to the doubly-virtual case, as required for the pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon.